Let $(M_{n})_{n}$ be an $\mathbb{F}$-Submartingale where $\mathbb{F}=(\mathcal{F}_{n})_{n\in \mathbb N_{0}}$and $\tau$ an $\mathbb{F}$-stopping time. I have shown that:
$E[M_{(n+1)\land \tau}\vert \mathcal{F}_{n}]\geq M_{n\land \tau}$ and now I need to prove that:
$E[M_{(n+1)\land \tau}\vert \mathcal{F}_{n\land \tau}]\geq M_{n\land \tau}$
My idea:
$E[M_{(n+1)\land \tau}\vert \mathcal{F}_{n\land \tau}]=E[E[M_{(n+1)\land \tau}\vert \mathcal{F}_{n}]\vert \mathcal{F}_{n\land \tau}]\geq E[M_{n\land \tau}\vert \mathcal{F}_{n\land \tau}]=M_{n\land \tau}$ since $M_{n\land \tau}$ is $\mathcal{F}_{n\land \tau}-$measurable
But I'm not sure whether I can use the tower property as above since I do not know whether it always holds that:
$\mathcal{F_{n\land \tau}}\subseteq \mathcal{F}_{n}$, is it true ?
If $\tau_1$ and $\tau_2$ are stopping times with $\tau_1 \leq \tau_2$ then $\mathcal F_{\tau_1} \subset \mathcal F_{\tau_2}$. I particular you can take $\tau_2=n$.